а ♡ 3 hoopdo not vanish. The cube roots in respect of the root unity are = w- and + N - 44; but if, under this conception, we should assimilate the sum of these roots to the root of the equation 3 x 28 = 2, a greater mistake, in my opinion, could not be made. In the same manner may my quantity be divided by 4, and it will be a cube root of unity, but never can it be conceived to be a root of the equation 3 x - del = 2; but if Dr. Tiarks's number be so xc3 = divided, it will be, together with 1-4, a root of the equation 3 - 3 * = 2. 3x . The equations 7 x = 6 and T 28 = 2 have a similar root unity; but it is seen that all equality is lost when they are converted into fractions, and this is precisely the case of our two numbers. It is the province of the lovers of the science to decide on the question. I am, Sir, your obedient servant, May 9, 1815. JAMES LOCKHART. - 1 t Another Communication on the same subject. 2 (To Dr. Thomson.) SIR, May 3, 1815. As the subject proposed by Mr. Lockhart on the algorithm of imaginary quantities is one of considerable importance in a variety of analytical investigations, you will be induced probably to admit a few remarks on the two answers published in your last number. The first thing which appears singular is, that one of your correspondents has shown Mr. Lockhart's expression to be the cube root of 64, but under a different form to that usually given; and the other, that it is not the square root of 64, but of 8. W3 The fact is, that Mr. L's expression, 3+=+ (* - 3), the same as all other quantities in which the sign of the square root enters, admits of two values ; and as there is no previous condition, either of them may be employed; and the quantity will be accordingly either the 64 or Ň 8. R. N. D. is therefore too positive when he says, " it is not the cube root of 64, but of 8.". He is also wrong in stating that by squaring : awb an ambiguity is introduced; for the ambiguity has place in the ✓ 1 before the operation of squaring ; in fact, the only case in which there is no ámbiguity is when we know the origin of the quantity whose root is to be extracted, as is shown in one of the latter numbers of Nicholson's Journal, where the object was to explain why Iv=+++ V =-3, which is 1 / IŅ- 3 t 1 / known to be equal to 1.87938, or 2 sin. 70°, is not (when squared by the usual process) equal to the square of the same number. The 3 , square of 3 + w 3 1 - 2 + 1 / ✓ 3) + 2, which is certainly not equal to 1.8793812 Query the cause of this anomaly, if we admit the ambiguity to arise as stated by your Correspondent R. N.D.? Yours, &c. MATHEMATICUS. ARTICLE IX. Astronomical and Magnetical Observations at Hackney Wick. By Col. Beaufoy. Latitude, 51° 32' 40•3" North. Longitude West in Time 6'1836 100 : 06" Ayril 188h 25' 24° 15' 33" 19:31 Ditto 22 8 40 24 16 54 Į45 24 25 46 Ditto 23 8 35 24 16 05 | 40 24 27" 06 6 25 24 19 02 Ditto 248 30 24 15 50 1925. 124 20715 Ditto 25 8 35 24 16 04 Ditto 28 8 35 24 17 32 105 24 28 49 Ditto 29 8 20 24 19 27 35 244 22 18 Ditto 30 8 20 24 16 30 195 24-1:29 21 10.50 24 18 50'y , May 1) gly 40' 24• 14' 50") 1b 5524° 29' 12'6h 50' 24° 20' 22" ; I 7 10 24 18 21 24 192 . April. The variation of the weather this month has been equalled by the uncommon variation of the needle. Rain fallen Between noon of the Ist May'} Apr.? 1:735 inch. Evaporation during the same period .1.90 Transactions of the Geological Society, Volume 2d. London, William Phillips, 1814. This volume contains 24 papers, and is accompanied by afvolume of plates. 1. On certain Products obtained in the Distillation of Wood, with some Account of Bituminous Substances, and Remarks on Coal. By J. Macculloch, M.D. F.L.S., Chemist to the Ordnance, and Lecturer on Chemistry at the Royal Military Academy at Woolwich, and Vice President of the Geological Society a When wood is exposed to destructive distillation there comes over a thick black fluid like tar. Muscar Great quantities of this substance are obtained at the powder works from the distillation of willow and alder. It was this substance that Dr. Macculloch examined. It is very inflammable, and may be burnt like oil in a lamp. When it is washed in water, that liquid separates a considerable proportion of acetic acid, coloured by an oily matter, which has an empyreumatic smell and taste. When triturated, or boiled with carbonate of potash, it acquires a pitchy consistence; but does not seem to combine fully with the alkali. It is soluble in alcohol, ether, caustic, fixed alkaline lees, acetic acid, and the mineral acids. Fat oils and fresh essential oils dissolve it imperfectly, but the drying oils and inspissated essential oils act on it inore readily. Coloured oil of turpentine dissolves a good deal of it. Naphtha has scarcely any action on it. When exposed to a heat just sufficient to keep it boiling, an oil comes over, at first light coloured, but becoming darker as the process advances. If the heat of the retort be gradually increased to redness, nothing remains but a spongy charcoal. There is found in the receiver an oil and acetic acid, combined with a little ammonia. No gas is evolved in this process, if the heat be carefully managed. When a gas is formed, the oil has been exposed to too bigh a temperature, by letting the fire act too much on the upper part of the retort. If the heat he continued for a certain time, what remains in the retort resembles. petroleum; if longer, maltha; if longer, bitumen; and if stilt longer, only coal remains. But these resemblances are merely external. Bitumen and our substance are different in their composition; since the first is soluble in naphta, the second insoluble. Dr. Macculloch conceives it probable that vegetables may have been converted into bitumen or jet, by the action of water, and the bitumen afterwards converted into coal by heat. He shows that heat is incapable of bituminizing wood, but that it converts jet into coal. Many other very ingenious hypotheses respecting the origin of the various species of coal and plumbago occur in this paper ; but they are of so bold a nature, and so little supported by the present state of our knowledge, that I am afraid to enter upon them, least they should lead to a tedious controversy about a subject, the decision of which is at present obviously beyond the reach of our faculties. The valuable part of the paper is the chemical description of the new substance obtained by distilling vegetables, which Dr. M. proposes to distinguish by the name of bistre. II. Mineralogical Account of the Isle of Man. By I. F. Berger, M.D., M.G.S. -The Isle of Man, the situation of which is too well known to require specification here, is rather more than 30 miles lung from north to south, while its breadth varies from 15 to eight miles. The northern extremity is tolerably low for about five miles, where a range of mountains commence that proceed to the southern extremity. This group of mountains consists of three certis he On ANETE chains, separated from each other by high table lands. Three very narrow openings cross the group from east to west. The furthest north of which lies between Douglas Town and Peel Town; and the furthest south is the narrow channel between the island and Calf of Man. The first of these is highest, and the last lowest, being under the level of the sea. The highest mountain in Man is Snowfield, which is 2000 feet above the level of the sea Dr. Berger has given the elevation of 89 mountains and spots of this island, which has enabled him to give interesting sections of the island in various directions. The Calf of Man is 206 feet high. The mean annual temperature he reckons from the heat of 81 springs, at 49.999, or 50°, Fahrenheit. Very little granite has been observed in Man. It occurs, however, in two or three spots; but most likely only forming beds in the clay-slate, which constitutes the central and highest part of the island. Grey-wacke covers the lower parts of this clay-slate , almost surrounds it, and constitutes the sea-coast both on the east and west side of Man. This grey-wacke sometimes contains beds of g grey-wacke slate and finty slate. It has not been observed to contain any animal remains. A floetz lime-stone containing madreporites, and some univalve shells, occupies the south-east coast; except a peninsula of old red sand-stone, which occurs likewise at Peel Town. The Calf of Man is grey-wacke. Veins containing galena occur in three different parts of the island, and always in the grey-wacke. Sulphuret of copper likewise occurs. These veins have been wrought more than once, ut are at present abandoned. Dr. Berger thinks that the southern part of the island has at some former period subsided. Hence he accounts for the gradual declivity, the dip, and the narrow valleys. The population is reckoned at 28,000; but Mr. Curwen thinks it does not exceed 23,000. III. On the Granite Tors of Cornwall. By Dr. Maeculloch. The Logging Rock, and Cheese Wring in Cornwall , and the Vixen Tor in Dartmoor, exhibiting curious configurations of granite rocks, are described and figured ; and Dr. Macculloch shows very clearly that they have resulted from the natural effects of the disintegration of the rock, and that they have not, as some supposed, been formed artificially. He terminates the paper with some speculations on the origin of granite, and rather inclines to its igneous origin. Into these speculations I think it unnecessary to enter ; because I copceive them to be entirely beyond the reach of the human faculties. Supposing a man to spend his life in guessing how the Almighty brought the earth into its present state, and supposing him even to guess right, (no very probable supposition) I ask by what means he could prove his conjecture ? The only conclusive evidence in such a case is historical evidence, and where could it be found? Geology does not consist in specylating about the origin or creation of the earth, it consists simply in determining the number, nature, and position of all the rocks |